2.2 State of the art of space–time measurements
2.2.3 Full spatio-temporal characterization techniques
Fourier Transform Spatio-Spectral Interferometry (FTSSI)
Fourier-transform spectral interferometry technique was presented in the previous chapter (section 1.5.2) as a strong referenced pulse characterization technique. Since its operation requires one dimensional array, adding the second dimension simply by implementing a 2D detector (CCD camera) and a 2D imaging configuration, can be used to measure the spatially resolved spectral phase of the pulses. This is what we
Figure 2.7: Top and side view of GRENOUILLE. A Cylindrical lens focuses vertically the input beam on to a biprism. The biprism divides the incident beam into two beamlets and combines them with an angle in the horizontal plane. A thick SHG crystal resolves and maps the spectral components to different angles in the vertical plane. An additional cylindrical lens at the other side of the crystal maps the angularly dispersed components on to different positions of the camera in the vertical plane. The signal at the camera is therefore a single-shot SHG FROG trace with delay running horizontally and wavelength running vertically. This figure is adopted from .
have used to spatio–temporally characterize the shaped pulses of the UV-AOPDF pulse shaper and I will present it in more detail in the following. Figure 2.8 (a) depicts a typical setup of a spatio-spectral interferometer in which the reference arm and the test pulse are combined collinearly with a relative delay at the entrance slit of a 2D imaging spectrometer. The imaging spectrometer selects one spatial slice of the overlapped beams. In this case, the spectrally resolved interfering electric field of different points along the beam slice yis recorded on different rows of the 2D camera.
Therefore, the recorded signal reads
S(ω, y) = |Eer(ω, y) +Ees(ω, y) exp(iωτ)|2 (2.1) which corresponds to spatially resolved spectral interference intensity [see Fig. 2.8 (b)]. The relative spatio-spectral phase can be extracted by using the Fourier-transform filtering algorithm for each point y of the interference pattern [see Fig. 2.8 (c)]. In
Figure 2.8: Schematic configuration of a collinear spatio-spectral interferometer along with the detected signal. (a) The test and reference arm are combined collinearly with a relative delay at the entrance slit of a 2D spectrometer. (b) The 2D spectrometer records the spatially resolved intensity interference of two arms. (c) The recorded interference fringes correspond to three lobes separated from each other by the amount of the relative delay τ in quasi-temporal domain.
order to reconstruct the spatio-spectral phase of the test pulse, the spatial and spec-tral phase of the reference arm should be already known. However, this task by itself is challenging because it requires a separate spatio-spectral pulse characterization technique. Having a reference pulse with spectral phase independent of its spatial transverse coordinate (space-time coupling free pulse) may ease the problem. In this case, by measuring the spectral phase at a chosen point y0 that contains all spectral components φr(ω, y0), one can obtain the spatio-spectral phase φr(ω, y) = φr(ω, y0) and therefore the spatio-spectrally characterized reference pulse. This consideration, a spatially independent spectral phase, maybe achieved by putting a pinhole in front of the test pulse, choosing a specific pointy0 from the wavefront of the test pulse and collimating it by means of a lens. However, there are at least two constraints for this approach. First, when the test pulse spectrum is ultra broad, it is almost impossible to make a collimated beam where all spectral components have a plane wavefront.
Secondly, when the test pulse is strongly spatio-spectrally coupled, it is difficult to find a position which contains all the spectral components. Nonetheless, spatially resolved spectral interferometry can be considered as one of the strongest tools that spatio-spectrally characterize the pulses. Its applications includes studying lens aber-rations [136,137], self focusing effects of transparent media  and spatio-temporal speckles .
Figure2.9 shows how FTSSI is capable of measuring different types of the space-time couplings. Figure2.9(a) shows the effect of the spatial chirp on the spatio-spectral in-terference pattern. It appears as a tilt of the elliptical spatio-spectral intensity pattern with respect to the spectral axis. Figure 2.9 (b) shows effect of pulse front tilt on the recorded interference pattern. The spectral fringe frequency δωf r ∝ 1/τ depends on the relative delay of two arms τ. In the presence of pulse front tilt along the entrance slit (vertical plane), the delay between two arms become position dependent. In more detail, each point along the slit in addition to fixed relative delay τ, also experiences different arriving times of the beams due to the tilt of the test arm with respect to the reference arm. The arrival time of this position dependent pulse tilts the interference fringe pattern, depending on the magnitude of the pulse front tilt, with respect to the vertical axis.
Figure 2.9: Spatially resolved interferograms of a spatio-spectral interferometer in the case of combination of a transform limited reference pulse with a spatio-temporally coupled test pulse. The presence of spatial chirp (a) tilts the elliptical spatio-spectral intensity pattern with respect to the spectral axis; while, the presence of the pulse front tilt (b) results in rotation of the interference fringe pattern. This figure is adopted from .
The spatio-spectral phase can be extracted by means of a Fourier filtering algorithm similar to the one dimensional spectral interferometry. However, as was mentioned in the presentation of Spectral Interferometry in the first chapter, using Fourier filtering
algorithms demands a spectrometer with higher resolution than it is normally required for sampling the finest features of the spectrum signal. A solution to this problem is crossing the arms on the entrance slit of the spectrometer with a small angle θ.
This results in appearance of the spatial interference pattern where its frequency is proportional toθ. Figure2.10(a) shows the recorded signal on the detector that reads S(ω, y) = |Eer(ω, y) +Ees(ω, y) exp(ikyy+iωτ)|2 (2.2) where,ky is the difference between transverse components of the propagation vectors k and related to crossing angleθ through
ky =ksinθ. (2.3)
Now, performing a two-dimensional Fourier transform along both the spectral and spatial axes yields three lobes. Here, in addition to the temporal separation the lobes are also separated along the quasi-spatial frequency dimension [see Fig. 2.10 (b)].
This means that the interfering term can be easily filtered without necessity of in-troduction of a relative delay. This approach therefore eases the spectral resolution requirement of the spectral interferometry. Depending on what the measurement
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t (fs) k y(rad)
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Figure 2.10: Homemade simulation of detected signal by a 2D spectrometer of a non-collinear spatio-spectral interferometer . (a) The 2D spectrometer records the spatially resolved interference of the transform limited reference and the test pulses with a rela-tive delay and angle. (b) The recorded interference fringes correspond to three lobes that are separated from each other by the amount of the relative delay τ and angle θ in the wavenumber-temporal domains.
quests, high spectral resolution or high spatial resolution, one can play with spatial and spectral resolution of the measurement by modifying the values of the relative
delay and relative angle. In more detail, the spectral resolution will be lost when the relative angle is zero and the interfering term is separated from the dc term just by relative delay or inverse, the spatial resolution is lost when the relative delay is zero.
To overcome the problem of compromise between spatial and spectral resolution of the FTSSI, SEA TADPOLE Spatial encoded arrangement for Temporal Analysis by Dispersing a Pair of Light E-fields, can be used [139,140]. In this method two pulses are combined by using of two monomode equal-length optical fibers [see Fig. 2.11].
In more detail, the reference arm is coupled into one of the fibers and the second fiber
Figure 2.11: Schematic configuration of the SEA-TADPOLE. A reference pulse and an unknown pulse are coupled into two single-mode fibers with approximately equal lengths.
At the other end of the fibers, the diverging beams are collimated using a spherical lens (f ).
After propagating a distance f, the collimated beams cross and interfere, and a camera is placed at this point to record the interference. In the other dimension, a grating and a cylindrical lens map wavelength onto the camera’s horizontal axis x. This figure is adopted from .
selects a specific point from the unknown pulse. Then the emerging beams from the fibers are crossed in the vertical plane with an angle which results in spatial fringes.
Leading the fringe pattern to the 2D imaging spectrometer results in spectrally re-solving the spatial fringes. In this case, the delay between two pulses are set to zero.
Since Fourier filtering algorithm filters the trace in the spatial coordinate, the spec-tral resolution of the measurement is kept untouched. Scanning the fiber transversely yields the spatially resolved spectral phase of the unknown pulse. Therefore, in this technique the spatial resolution is defined by the mode size of the fiber. SEA
TAD-POLE is a strong measurement tool for measurement of the spatio-spectral phase of the tightly focused pulse giving that the mode size of the fiber is several times smaller than the size of the focus. However the challenging issue concerning this technique is the calibration of the relative delay between the two arms, because transversely scanning the fiber also introduces longitudinal fluctuation.
All above mentioned devices are referenced measurement techniques. Similar to Spec-tral interferometry that is extended to measure the additional spatial domain by using of a 2D imaging spectrometer, spectral shearing interferometry also can be extended to spatio-spectral phase measurement device. SEA-SPIDER, Spatial Encoded Ar-rangement SPIDER  is the name of the developed device for spatio-spectral measurement of the pulses. In this technique, a 2D imaging spectrometer is im-plemented instead of non imaging spectrometer. Moreover, two spectrally sheared replicas, instead of being temporally delayed, are combined with an angle in to the imaging spectrometer. In this case, spatial fringes are produced instead of spectral fringes that relax the spectral resolution of the spectrometer through the Fourier fil-tering algorithm as explained before. Furthermore, using an imaging spectrometer gives the possibility of measuring the spectral phase at each point along the unknown pulse wavefront. However, SEA-SPIDER is a self referenced technique that does not retrieve the group delay of the unknown pulse. This means that the spatially depen-dent group delay of the pulse (spatial phase) remains unknown. Nonetheless, it reveals interesting signatures of spatio-spectral coupling of the pulses. One can encompass this problem by applying both spatial and spectral shearing simultaneously. This is called 2D shearing interferometry  in which two spectrally sheared replicas of the pulse are combined by a small angle with additional spatial displacement. Spatial and spectral phase can be separately extracted via Fourier filtering algorithm and then stitched together to retrieve a complete spatio-spectral characterization of the pulse.
So far, because of applying 2D detectors, all mentioned spatio-spectral characteri-zation techniques are limited at best to measure spatio-spectral phase of the pulses φ(y, ω) where the information about the other spatial dimension is lost. Fortunately, in most cases the generated spatio-spectral couplings occur only between one spa-tial and spectral (temporal) dependence of the pulse electric field. Therefore, the
loss of one spatial dimension is acceptable. However, some fields require the whole spatio-spectral characterization of the pulses φ(x, y, ω). 2D SPIDER and SSI tech-nique can adopt themselves to 3D measurement by scanning the entrance slit of the spectrometer. This requires a multishot measurement which is not always desired due to its several constraints. A different approach is STRIPED-FISH [142,143], a form of Fourier-transform interferometry with a spatial carrier. In this technique, the test and reference beams pass through a coarse grating, which produces many diffracted orders. These diffracted orders then pass through an interference filter.
Since each order has a slightly different wave-vector and the passband of the filter is angularly dependent, the passband frequency seen by each order is different. Each order therefore contains a quasi-monochromatic spatially encoded interferogram, and these are simultaneously recorded on a two-dimensional detector. Combining the in-terferograms yields a three-dimensional dataset. However the spectral resolution of the reconstructed electric field is reduced.